Jeffrey E. Barrett
Illinois State University
Douglas H. Clements
State University of New York at Buffalo
Children coordinate their numerical and spatial knowledge whenever they measure space in two-dimensions. In this teaching experiment, fourth-graders measured and described general cases of polygons that satisfied specific constraints for perimeter or side length. Findings suggest that the transition between uni-dimensional and bi-dimensional geometric contexts requires extensive coordination operations involving numerical and spatial knowledge structures if children are to establish robust structures linking length, perimeter and area concepts. The progress of one child is examined in the context to elaborate on the ways he restructured his strategic knowledge of length to incorporate measures of perimeter. Conclusions regarding instructional sequencing are given, along with suggestions for continuing investigation.
Curriculum in geometry typically assumes that children can lean to distinguish area from perimeter by fourth grade, although national tests have indicated otherwise (Lindquist & Kouba, 1989). As noted by Kamii, many educators suggest that children usually confuse area and perimeter concepts, leading to erroneous responses on both types of task. Kamii counters that children make errors on tasks involving area because of their inability to abstract area from linear measures of space; children cannot understand area until they gain the abstract notion of area as continuous 'band' or matrix of one-dimensional lines (1996). Even the measure of linear quantity along a line segment is problematic for many children younger than twelve years; children who have not developed iterative strategies are inclined to count items rather than sub-segments, overlap or gap their iterative operations, and miscount the endpoints, especially with ruler markings (Boulton-Lewis, Wilss, & Mutch, 1996; Cannon, 1992). Thus, perimeter tasks, like area tasks, may demand extensive coordination of linear operations (Barrett & Clements, 1996). We argue here that operations for measuring perimeter cannot be interpreted as counting operations within a single dimension, but as a coordination of one-dimensional objects within two-dimensional space.
This paper is part of a wider study of children's understanding of length concepts; we carried out a comprehensive teaching experiment with four children during the Spring of 1997 (Barrett, 1998). These children represented distinct strategies for length and measurement. We sought to characterize the mathematics of the children themselves, attempting to make sense of their strategies and progress from within their own collection of concepts and practices (Steffe & Thompson, in press). Our analysis was two-fold, involving an on-going negotiation between the mathematical thinking of the subjects and our developing models for their geometrical thinking. We employed a framework developed by Clements et. Al. (1997) and a set of constructs that derived from a pilot study (Barrett & Clements, 1996). We focused intensively on negotiations, conflict meanings and growth as indicators of developing abstraction through increasingly connected representational structures.
Setting a Context for Alex's Story
Anna, Alex, Natasha and Paul exhibited four distinctive strategies for length. We employed a set of three strategy levels to analyze their work (Clements, 1997). Alex moved from an early level 2 strategy to a more advanced level 2, verging on the use of level 3 strategies near the end of the study. We begin our account of Alex's developmental "growth spurt" by summarizing the changes in the other three. First, Anna, understood length by direct, visual, gross comparison, and sometimes by making inexact correspondence between her counting sequence and the number of visible markers along an object. Anna eventually came to depend less on direct comparison, reflecting instead on her motor activity. Paul, in contrast, understood length by a formalistic and abstract process; he projected conceptual units of units in one- and two-dimensional units of measure. Alex and Natasha tended to express length as the number of perceptual markers counted along an object. They counted pseudo-units of length. While Alex often employed visual guessing and did not establish consistent counts between parts and whole for polygons, Natasha was more consistent in her use of hash marks for length and perimeter. In particular, Natasha was able to establish a careful correspondence along a single dimension, but not along a two-dimensional path, whereas Alex usually failed to establish such correspondence. We sought to understand Alex's attempts to resolve perturbations he met while trying to coordinate his counting sequences along one-dimension with his active re-presentation of two-dimensional perimeter tasks.
Counting discrete items along an object: An inadequate strategy
In the first session Alex measured length along an object by pointing to the visible markers sub-dividing it; he counted along a plastic straw 48 cm in length, marked at 2 cm intervals with small notches cut from its surface. The interviewer suggested that the straw was 24 long, showing him a way of sectioning off first one part of the straw by bending it at the first notch, and then three, bending it at the third notch. Then he asked Alex to find the length:
|Alex:||One, two, three, . . . , twenty-two, twenty-three.|
|Int.:||Okay, and you are touching what?|
|Int.:||And you are counting what?|
|Alex:||How much there are.|
|Int.:||How much what? 23 or 24 what?|
|Alex:||That there's a length.|
Alex believed he had found the length by counting the number of holes along the edge of the plastic straw. Initially, he appeared to take the holes as components of the length along the straw. Alex did not appear to operate on a conceptual image of units of length. His way of marking length did not include the generalization that would have led him to point finally to the end of the straw: he always stopped at the last hole, failing to count the last interval along the straw. However, later during that same interview, Alex drew a rectangular figure that he marked by making both hash marks and dots, placing a dot in between each set of harsh marks, 24 dots in all around the perimeter of the rectangular figure. The dots corresponded exactly with the partitions created by the hash marks. Thus, by the end of the first interview Alex was able to discriminate between marks and delimited subdivisions along a continuous line segment (indicated by the hash marks) and the subdivided portions of the line segments (indicated by the dots). Nonetheless, Alex persisted in stating that the straw was 23 long, and not 24.
Apparently Alex allowed a disjointedness between his perceptual images for length and his conceptual notions for iterative counting operations on length tasks, much in the same way that the children in Fischbein's study could at once explain that a geometric 'point' might not constitute length or area in one context, yet attribute length to that point in a different figural context (1993). Alex needed an operational model for connecting one-dimensional segments into perimeter.
System such as this can help establish connections among related ideas, both within mathematics and across other subject matter areas. Systems that generate an array of interdisciplinary problems and track multiple concepts and processes may facilitate the development comprehensive interdisciplinary instruction (Lipson et al., 1990).
Abstracting a 'wrapping' metaphor for perimeter: putting fringe on a rug
By now, Alex had used tiles in two different ways by this time during the teaching experiment, both to find length and distance across a band of tiles, and as a way of covering a two-dimensional region. During the third session, Alex combined his existing scheme for length with a 'wrapping' scheme: The interviewer asked him to consider how many "tiles" worth of fringe one would have if they were to wrap fringe around a rectangular-shaped rug in that room (the rug was roughly 2 tiles wide and three tiles long). He eventually found that it would require about 10 tiles worth of string. Later, the interviewer asked him to imagine placing fringe materials around a rug that would be seven tiles by six tiles on its sides. Alex looked down at the pattern of floor tiles. He started to walk and count aloud as before, taking four steps in the first four tiles, but stopped:
|Int.:||Tell me about the perimeter of this rug. How much fringe do you need to buy this rug?|
|Alex:||[starts to walk around it, taking four steps inside the tiles but suddenly he stops, pauses, and begins talking:] 14, 12, ... [inaudible words here] 6 and 6 is 12 --- 7 and 7 is 14 --- 10 and 10 is 20 --- plus 4 plus 2 is 6, --- it's 26.|
|Int.:||You got 26?|
He was beginning to depend on arithmetic schemes for numbers: he began trying to compose perimeter by summing two pair of equivalent values, based on symmetries (e.g. "5 and 5 is 10"). While this spontaneous reorganization led him to stop walking around the physical perimeter (the rectangular figure in the tile floor), as the interviewer, I could not yet be sure of what kind of length units he intended. I suppose his invention of an arithmetic solution was not yet integrated into his figurative perception of the floor tile pattern. So I asked him about his meaning for "units". He offered a tentative response that it was "26 tiles". At this point he used large sweeping motions of his arms to point out entire rows and columns of tiles in the floor along the four edges:
|Int.:||What does 26 mean here?|
|Int.:||Are there 26 square tiles? Where are they?|
|Alex:||Seven across here [sweeping his hand along one edge], and six across there, [sweeping his hand along the next edge] and 7 across there ...|
At the time, I believed he was referring to the literal tiles, both and columns of tiles (suggesting an area unit). His hand-sweeping motion toward rows and columns of tiles confirmed this belief.
As he stood looking at the tile floor and trying to describe how the 26 tiles could make up the perimeter of the rectangle patter, I asked him once more to identify the 26 tiles. I expected that he would either count each corner tile once and find only 22 tiles, or count outside the figure and find 30 tiles. He began to reflect on his own actions and on his efforts to coordinate four sides:
|Int.:||So show me the 26 tiles. Where are they? Can you step in one at a time and show me all 26?|
[Alex walked around two sides and counted aloud up to 13, but seemed concerned. He halted immediately after stepping through the second side of the figure. I asked him to start over for the camera:]
|Alex:||One, 2, 3, 4, 5, 6, 7, [7 is the corner tile. He turns the corner, and says:] 8, but no, it can't be, (with unusual vocal emphasis) [pausing]|
|Int.:||What do you mean?|
|Alex:||Because you gotta add an extra tile, or use it again.|
|Alex:||'Cause I already used this tile and I gotta use it again.|
|Alex:||Because I will not get six unless I do.|
Alex paused. He was still counting the corner tile as one unit of length, and so by the time he reached the final tile on this second side (6 tiles long), he had only counted on by five, reaching twelve, but he said "thirteen". His hesitation was apparently based on reaching 13 unexpectedly.
Alex was ready to re-organize his tile counting scheme for length, but only when his scheme kept him form counting around corner; he wanted to count the corner tile twice now, but lacked an effective justification for his new scheme. He needed a conceptual scheme to fit this context; Alex was still constrained by his one-dimensional concept of perimeter, following from his use of arithmetic operations as connectors to extend his one-dimensional measures of length into two-dimensional world of perimeters and polygons. Kamii (1996) suggests the need for children to distinguish between discrete quantities of square tiles and continuous regions, and of distinguishing between uni-dimensional thinking and bi-dimensional thinking respecting area tasks. Cannon (1992) distinguished between counting discrete symbols directly as items in a collection and counting symbols indirectly as subdivisions along a continuous dimension. We argue here that the operations necessary for measuring perimeter cannot be interpreted as counting along a given dimension, but as a coordination of one-dimensional objects through a second dimension.
All four children in the study were distracted by the perceptual salience of the notches on the straw, and by the square tile images; they struggle to isolate length along edges in these complex settings (Steffe, 1991). For example, Alex learned, through the course of these sessions, to differentiate between abstract definitions (which Fischbein terms "conceptual geometric knowledge") and figural images (which Fischbein terms "figural geometric knowledge"). When he met crises, like when his arithmetic knowledge led him to question his counting of just 12 tiles along two edges that he expected to sum to 13, Alex's abstract numerical notion took precedence over his figural images, a finding consistent with work by Ferrari (1992). This account of Alex's work supports the generalization that children who integrate their conceptual and figural-imagistic knowledge resolve perturbations more easily than those who do not (Clements, 1997).
How children learn to distinguish between continuous and discrete quantity? In this study, Alex often tried to coordinate his experienced iterations along a linear object and the reified markers along that object symbolizing a previous movement through continuous space. Whenever these two sequences did not correspond exactly, there was a perturbation involving the symbolized unit (a continuous unit) and the symbolic item (a discrete item). Alex was forced to distinguish between parts of continuous quantity (the un-segmented linear object) and the delimiters used to partition that quantity (the hash marks) (Steffe, 1991). The question, "what are children counting for length?" still needs to be addressed in further detail. The question can be posed more generally: "how does one count a continuous quantity?" Ultimately, measurement operations appear to undergrad children's knowledge of quantity and counting.
Barrett, J.E. (1998). Representing, connecting and restructuring knowledge: The growth of children's understanding of length in two-dimensional space. Unpublished Dissertation, State University of New York at Buffalo, Buffalo.
Barrett, J.E. & Clements, D. H. (1996, October 12-15). Representing, connecting and restructuring knowledge: A microgenetic analysis of a child's learning in an open-ended task involving perimeter, paths, and polygons. Paper presented at the Eighteenth Annual Meeting: North American Chapter of the International Group for the Psychology of Mathematics Education, Panama City, Florida, U.S.A.
Boulton-Lewis, G. M., Wilss, L.A., & Mutch, S.L. (1996). An analysis of Young Children's Strategies and Use of devices for length measurement. Journal of Mathematics Behavior, 15, 329-347.
Cannon, P.L. (1992) Middle grade students' representation of linear units. In W. Geesling & K. Graham (Eds.), Proceedings of the Sixteenth PME Conference (Vol. 1) (pp. 105-112). Durham, NH: Program Committee of the 16th PME Conference.
Clements, D. H., Battista, Michael T., Sarama, J., Swaminathan, S., McMillen, S. (1997). Students' development of length measurement concepts in a Logo-based unit on geometric paths. Journal for Research in Mathematics Education, 28(1), 49-70.
Ferrari, P.L. (1992). Problem-solving in geometrical setting: Interactions between figure and strategy. In W. Geeslin & K. Graham (Eds.), Proceedings of the Sixteenth PME Conference (Vol. 1) (pp. 217-224). Durham, NH: Program Committee of the 16th PME Conference
Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139-163.
Kamii, C. (1996, October 12-15). Why can't fourth graders calculate the area of a rectangle? Paper presented at the Eighteenth Annual Meeting: North American Chapter of the International Group for the Psychology of Mathematics Education, Panama City.
Lindquist, M. M., & Kouba, V. L. (1989). Geometry, In M. M. Lindquist (Ed.), Results from the Fourth Mathematics Assessment of the National Assessment of Educational Progress (pp. 44-54). Reston, VA: National Council of Teaching of Mathematics.
Steffe, L., & Thompson, P. (in press). Teaching Experiment Methodology: Underlying Principles and Essential Elements. In R. Lesh & E. Kelley (eds.), Research design.
Steffe, L. P. (1991). Operations that generate quantity. Learning and Individual Differences, 3(1), 61-82.
Other Articles by Douglas H. Clements: